PERIODIC MOTION 331 When the contact is smooth we have T= Jm(&-a)202, and U = J mg (b - a) 02. Therefore 6 = a sin \ /T-2— (« + *0) , V 6 — a and P = 2 Thus the length of the equivalent simple pendulum is (6 — a) when the contact is smooth and — — - — - when it is rough. PROBLEMS. 1. A butcher's balance is elongated 1 inch when a weight of 4 pounds is placed in the pan. If the spring of the balance weighs 5 ounces, find the error introduced by neglecting the mass of the spring in calculating the period of oscillation. 2. Find the expression for the period of vibration of mercury in a J7-tube. 3. If in the illustrative problem on p. 329 the particle divides the string in the ratio of 1 to n, show that the period is P = 2 Try n g • ~^- - 4. Find the period of vibration of a homogeneous hemisphere which performs small oscillations upon a horizontal plane which is rough enough to prevent sliding. 6. Find the period of vibration of a homogeneous sphere which makes small oscillations in a fixed rough sphere. 6. A particle of mass m is attached to a point on a smooth horizontal table by means of a spring of natural length L. If the particle is pulled so that the spring is stretched to twice its natural length and then let go, show that it will vibrate with a period P = 2 (TT + 2) y~-, where T is the force necessary to stretch the spring to twice its natural length. The mass of the spring is negligible. 7. Two masses mi and m2 are connected by a spring of negligible mass. The modulus of elasticity of the spring is such that when mL is fixed m2 makes n vibrations per second. Show that when m2 is fixed mi makes n V ~ vibrations per second. v wi